Attention Manipulation and Information Overload:Barriers to Consumer Protection⇤

Petra Persson

†

March 2016

Abstract

Consumers’ attention limitations give firms incentives to manipulate prospec-tive buyers’ allocation of attention. This paper models such attention manip-ulation and shows that it limits the ability of disclosure regulation to improveconsumer welfare. Competitive information supply, from firms competing forattention, can reduce consumers’ knowledge by causing information overload.A single firm subjected to a disclosure mandate may deliberately induce suchinformation overload to obfuscate financially relevant information, or engage inproduct complexification to bound consumers’ financial literacy. Thus, disclo-sure rules that would be welfare improving for agents without attention limi-tations can prove impotent for consumers with limited attention. Obfuscationsuggests a role for rules that mandate not only the content but also the formatof disclosure; however, even rules that mandate “easy-to-understand” formatscan be ineffective against complexification, which may call for regulation ofproduct design.

Keywords: Information Overload, Complexity, Rational Inattention, Per-suasion, Disclosure regulation, Consumer Protection, Salience

JEL Codes: D11, D14, D18, D83

⇤I am grateful to Doug Bernheim, Patrick Bolton, Yeon-Koo Che, Pierre-André Chiappori,Takakazu Honryo, Navin Kartik, Samuel Lee, Uliana Loginova, Florian Scheuer, Cass Sunstein, andto seminar participants at the Consumer Financial Protection Bureau and at various universities.I also thank the faculty and participants of the Russell Sage Foundation Summer Institute inBehavioral Economics.

†Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305. Email:perssonp@stanford.edu.

1 Introduction

Governments can increase welfare by addressing market failures and providing publicgoods. In addition, governments often take actions to make constituents betterinformed, hoping to raise welfare by improving consumer decisions. Governmentalagencies such as the Food and Drug Association (FDA) and the Consumer ProtectionAgency (CPA) disseminate recommendations and guidelines. In addition, firms areincreasingly mandated to disclose information that is deemed relevant to consumers.Credit card companies must disclose important details of the products they offer,and food producers must adhere to labeling regulations, for example.

But processing this information requires consumers’ attention. If attention isunlimited, more information is weakly better, so regulation that mandates disclo-sure cannot harm. There is growing evidence that consumers’ attention is limited,however (see, e.g., Chetty et al. (2009); Dellavigna and Pollet (2009); Abaluck andGruber (2011)). Then, the welfare consequences of regulation that mandates dis-closure are, a priori, less clear, as consumer attention devoted to one piece of infor-mation may crowd out attention to others. The literature on rational inattentionanalyzes how a decision-maker with limited attention allocates it among various pas-sive sources of information (e.g., Sims, 2003; Wiederholt, 2010). The core idea inthis paper is that consumers’ limited attention gives information providers, such asfirms, incentives to be active. To be more precise, when a consumer’s attention islimited, her ultimate purchasing decisions may hinge on what she pays attentionto; this, in turn, gives firms incentives to engage in attention manipulation, that is,strategic actions to influence how she allocates her attention. This is distinct from,and operates on top of, any incentive to manipulate the substance of communication.

To the best of my knowledge, this paper is the first to propose and analyze at-tention manipulation. The first part of the paper shows that, in the presence ofattention manipulation, competitive information supply, from firms competing forattention, can reduce the consumer’s knowledge by causing information overload.The second part shows that a single information provider, such as a firm mandatedto disclose information, may deliberately induce information overload to conceal in-formation. Thus, requiring a firm to disclose all hidden, undesirable features of itsproduct may have no impact on social welfare; the firm will simply disclose these fea-tures along with an avalanche of irrelevant information. But if full disclosure policiescan backfire, intuition suggests that there is an easy fix: to simply mandate not only

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what firms disclose, but also how it should be disclosed. The paper’s third result isa disconcerting one, however: Mandating firms to provide easy-to-understand infor-mation about the products they sell will induce “complexification” of the underlyingproducts themselves. This is essential as it limits the potential welfare gains frommandated information provision; in fact, complexification can eradicate all welfaregains. Together, these findings demonstrate that taking attention manipulation intoaccount have important implications for the design of consumer protection regula-tion, by suggesting a role for rules that restrict communication, mandate not onlythe content but also the format of disclosure, and regulate product design.

These findings arise in a theoretical framework of information exchange betweenone consumer (decision-maker; henceforth “DM”) and various suppliers of informa-tion (firms; henceforth “experts”). To capture attention manipulation, I want to per-mit the experts to be active, in the sense that they make persuasion effort choicesthat, in turn, affect the optimal attention allocation of the DM. I depart from theframework by Dewatripont and Tirole (2005), who model a DM interacting with asingle expert. The DM considers taking an action that has an uncertain payoff toher, but that would benefit the expert for sure (e.g., buying a good from the expert).Before the DM decides, she can communicate with the expert. Communication isa moral-hazard-in-team problem: The more attention the DM pays, and the moreeffort the expert makes, the more likely that information is exchanged successfullybetween them. Dewatripont and Tirole (2005) refer to this as issue-relevant commu-nication, because it concerns the DM’s actual benefits from the action. Importantly,the expert does not know what conclusion the DM will draw from the information heprovides; he only knows that it may affect her decision.1 In addition, Dewatripontand Tirole (2005) allow for a pre-play stage in which cue communication takes place.It does not concern the actual benefits associated with the action, but rather thedecision’s ex ante appeal, that is, the likelihood that (issue-relevant communicationwill show that) the action is beneficial.

1Intuitively, a firm representative selling a product can expend effort to convey more informationto the buyer about the product, without knowing with certainty whether it will make the prospec-tive client conclude that her payoff from the product is positive (and choose to buy) or negative(and choose to abstain). Thus, information provided from the seller is always truthful in this set-ting; put differently, no vendors are behaving in an unlawful manner – their choice regarding howmuch persuasion effort to make simply concerns how much effort to make to clearly and truthfullyconvey additional information about the product to the buyer. As we shall see, even though all

communication is required to be truthful, this does not guarantee that a seller lacks strategies toeffectively hide information that he deems would reduce the consumer’s willingness to purchase.

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To analyze attention manipulation, I introduce multitasking into this framework.A single DM (she) considers several binary actions, and can communicate with onedistinct expert (he) on each of them. Each of these issue-relevant information ex-changes is a moral-hazard-in-teams problem, and the DM now faces a multitaskingproblem because she must divide her limited attention between the various experts.In the first part of Section 2, I show that, in this framework, attention substitutionleads to externalities that I refer to as attention crowding out : Each expert ignoresthat his persuasion effort choice affects the attention that the DM devotes to otherexperts. Interestingly, an expert benefits or suffers from crowding out, dependingon whether he believes that attention from the DM will raise or lower the likelihoodthat she takes the action that he benefits from. This makes an expert’s expectedpayoff non-monotonic in the appeal of other experts’ proposed actions.

In the second part of Section 2, I allow each action’s ex ante appeal to be un-observed to the DM, and add a stage before issue-relevant communication, in whichcue communication can take place. Specifically, each expert can, at a cost, send hardinformation about his proposed action’s appeal to the DM, and she can process thiscue at a cost. Intuitively, in the presence of competition between experts for theDM’s attention, cue communication takes place first and helps the DM select whichexperts (actions) to devote attention to in the second stage. Cue communicationthus shapes the set of actions that the DM ends up deliberating on. For this reason,I say that cue communication takes place in the “selection stage” and issue-relevantcommunication in the subsequent “deliberation stage”.

I analyze how the DM’s welfare changes as the cost of sending cues—or proposingactions to the DM—falls and, consequently, more experts seek attention and morechoices enter the picture. Initially, she benefits from the fact that she has moreactions to choose from. But as entry becomes cheaper, it becomes profitable forexperts who propose actions with a lesser appeal to enter. As a result, the averagequality of the proposed actions deteriorates, and the DM must read cues, at a cost,to find the attractive ones. Eventually, as the supply escalates, screening ceases tobe worthwhile for her, and she picks random proposed actions for deliberation.

Thus, at a certain point, as the competition for the DM’s attention increases andshe gets more information, she processes less of it—or tunes out—and fares worse.I refer to this as information overload. Its immediate cause is that the quality ofthe proposed actions decreases with the quantity; actions worthy of deliberationbecome the proverbial needle in a haystack. The deeper cause, though, is negative

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externalities: Entry is individually rational for each expert, even as it complicatesthe selection problem for the DM and spoils overall communication. A DM withlimited attention may hence want to limit access to her attention space, even ifthat reduces her choice set. She faces a trade-off between comprehensiveness andcomprehensibility.

Section 3 reinterprets the framework to capture a DM who communicates witha single expert on one action that has several aspects. In the deliberation stage, theDM’s multi-tasking problem now stems from the fact that the DM must decide howto allocate her scarce attention between the various aspects of the action. In theselection stage, recall that in the “multiple experts”-setting analyzed in Section 2,the DM is unsure of each action’s ex ante appeal. The analogue in Section 3 is thatthe DM is unsure which aspects of the (single) action are financially relevant, andhence worth devoting attention to in the deliberation stage.

The analysis in Section 3 picks up on the information overload result from Sec-tion 2 and shows that a single expert may induce this outcome. Specifically, inSection 2, information overload resulted from various experts’ competition for atten-tion. Section 3 shows that a single expert, who faces no competition for the DM’sattention but who can communicate with the DM about multiple aspects of the ac-tion he advocates for, may strategically induce information overload in the DM. Thisarises when the expert wants to divert attention away from relevant aspects that are“unfavorable” in the sense that, if the DM learns more about those aspects, it maypush her to not take the action. Of course, if up to the expert, he would not bringup any such unfavorable aspect – that is, he would not send any cue about it. Butsometimes the expert cannot or may not withhold such information, e.g. due to lawsthat mandate disclosure of relevant information. I show that, in response to sucha disclosure mandate, the expert chooses to inundate the DM with cues relating tomostly irrelevant aspects of the action. This induces information overload, which,in turn, effectively conceals the inconvenient aspects that the firm was mandated todisclose in the first place. In other words, the expert shares superfluous informationto strategically induce information overload, which essentially obfuscates the DM.When consumers’ attention is limited, simple disclosure rules can thus be completelyimpotent and have no impact on social welfare.

But if disclosure mandates can backfire because firms can induce informationoverload, intuition suggests an immediate solution: to mandate not only what firmsdisclose, but also that it is disclosed in an easy-to-understand way. The second part

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of Section 3, however, shows that mandating a firm to provide easy-to-understandinformation about the product it sells can induce a “complexification” of the under-lying product itself. I show this by extending the model with a single expert to allowthe DM’s payoff from the (single) action to be comprised of many components, andto allow the expert to manipulate that composition so long as the total payoff staysconstant. Intuitively, such payoff-equivalent variations amount to changing the num-ber of financially relevant aspects of the product. This gives the expert yet anothertactic to thwart learning: The expert can force the DM to understand more details ofthe action, or product, to grasp its total payoff; in other words, he can make it morecomplex. Complexification ensures that an increasing amount of relevant informationslips the DM’s attention and, by the same token, that whatever she can learn in thedeliberation stage is so trivial that it no longer affects her decision. In a nutshell,even if she fully understands all the aspects that she can manage to deliberate on,she will always do what she would have done anyway. Complexification thus hassame welfare consequences as inducing information overload — it can eradicate allintended welfare gains from mandated information provision. But complexity is amore delicate issue for regulation: Unlike strategic information overload, it cannotbe tackled at the level of communication, information, and disclosure; it may call forintervention in product design.

To my knowledge, this paper presents the first economic model of attention ma-nipulation. It builds on and contributes to several strands of the literature. First,I advance the recent work on two-sided communication as a moral-hazard-in-teamsproblem, where the “softness” of information is intermediate and endogenous (De-watripont and Tirole, 2005), by introducing multiple experts that vie for the DM’sattention.2 Competition for attention leads to attention substitution, which in turninvites attention manipulation. More generally, this relates to a number of studiesthat examine how a DM communicates with multiple experts or with a single experton multiple topics. Krishna and Morgan (2001), Battaglini (2002), and Ambrus andTakahashi (2008) study competing experts in a soft information setting. In contrast,Milgrom and Roberts (1986) and Gentzkow and Kamenica (2011) study competing

2Soft information can be misrepresented at no cost (Crawford and Sobel, 1982); hard informationcan be withheld but not misrepresented (Grossman, 1981;Milgrom, 1981). When communicationis a moral-hazard-in-teams problem, the softness is intermediate: Communication conveys hardinformation with a probability that depends on effort by both sides; otherwise, information remainssoft. Introducing a lying cost represents another way to bridge soft and hard information (see, e.g.,Kartik et al., 2007; Kartik, 2009). Also see Caillaud and Tirole (2007).

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experts in a hard information setting, and Chakraborty and Harbaugh (2007, 2010)study soft communication between a DM and one expert on several topics. In thesepapers, competition or multiplicity never reduces the amount of knowledge the DMgains. Key to the result in my paper that more information can reduce the DM’sknowledge is that experts manipulate not only the substance of communication butalso the DM’s attention allocation. This suggests that limits to attention are im-portant for whether individuals stand to benefit from more competitive, or greater,information supply.3

Second, the two key results in Section 3 relate to Carlin (2009), Wilson (2010),and Ellison and Wolitzky (2012), papers that model obfuscation or complexificationas a strategic choice by firms to raise search costs in settings with optimal consumersearch.4 In these papers, firms’ incentives to raise these search costs stem fromcompetition between firms. This paper shows that obfuscation and complexificationcan be individually rational even when a single firm operates in the absence ofcompetition, if it is subjected to disclosure regulation. In addition, the currentpaper makes a distinction between obfuscation – which alters communication abouta product without altering the product itself – and complexification – which affectsthe product characteristics per se. While their consumer welfare consequences aresimilar, the distinction has crucial implications for optimal disclosure regulation.

Third, Bordalo et al. (2012, 2013) model to which attributes an individual’sattention is drawn when it is limited: attention is unproportionally allocated tosalient issues. I instead focus on how, when individuals have limited attention,market participants’ strategically take actions to make a certain attribute of a goodsalient or invisible, depending on whether the market participant wants to concealor emphasize the attribute. Put differently, I allow interested parties to influencethe relative salience of a product’s attributes. This relates to Bordalo et al. (2015),who study a competitive market where the salience of goods’ characteristics areendogenously determined.

3In these models, all the experts have information relevant to the same action. In contrast, Ianalyze a case in which each expert has information about a different action. This relates to a liter-ature in organizational design in which multiple division managers communicate local informationto a central management (Dessein and Santos, 2006; Alonso et al., 2008). This literature, however,deals with neither limited attention on part of the DM nor competition for such.

4In settings where multiple firms sell a homogenous good and compete on price, Wilson (2010)and Ellison and Wolitzky (2012) ask why it is individually rational for firms to raise consumersearch costs, and Carlin (2009) focuses on how this affects market prices. Other papers that discussobfuscation mechanisms in the context of competitive price discrimination models include Ellison(2005), Gabaix and Laibson (2006) and Spiegler (2006).

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2 Many Experts Competing for Limited Attention

In this section, I introduce the multiple expert setting and analyze, in turn, thedeliberation stage and the selection stage.

2.1 The deliberation stage

Set-up I introduce multiple senders, or experts, into the framework proposed byDewatripont and Tirole (2005). A DM faces two simultaneous decisions, i = 1, 2.

5

Each decision i concerns whether to take a distinct action, Ai. For each decision,there is a distinct expert who gets a deterministic payoff d > 0 if the DM takesthe action, and zero otherwise. The DM’s payoff xi from Ai takes the value x > 0

with probability ↵i and otherwise the value x < 0. The probability ↵i is commonknowledge. The larger the ↵i, the more attractive Ai seems to the DM, and themore aligned are her interests with those of the expert vested in Ai. Everyone isrisk-neutral. In the absence of additional information, the DM takes Ai if and onlyif its expected payoff is positive: ↵i > ↵

⇤ ⌘ �xx�x .6

Before making any decision, the DM can learn more about the actions. Foreach action, the vested expert can provide information, and the DM can devoteattention to processing this information. Through such communication, the DMcan learn the realization of xi. The expert himself knows neither whether xi =x orxi = x nor whether the (truthful) information he provides will help the DM findout. Nevertheless, his information may persuade the DM to take Ai even though↵i < ↵

⇤, since the DM may find out that xi=x. The probability that the DMlearns xi is given by p (si, ri), where si and ri are, respectively, the expert’s effort tocommunicate about Ai and the attention that the DM devotes to learning about Ai.

Assumption The function p (si, ri) is twice continuously differentiable on [0, 1]

2,with p (0, 0) = 0 and p (1, 1) = 1. It is strictly concave and satisfies p1(·) > 0,p2(·) > 0, p12(·) > 0, and the Inada condition 8 si 2 [0, 1], p2(·) ! 0 as ri ! 1 and

5It is not essential that the decisions are simultaneous; only that communication about bothdecisions is simultaneous.

6W.l.o.g., I assume that she does not choose A

i

when ↵

i

= ↵

⇤. Dewatripont and Tirole (2005)refer to this as supervisory decision-making, which they distinguish from executive decision-making,whereby the DM chooses action A

i

only if she is certain that xi

= x. Intuitively, executive decision-making may capture the DM’s behavior when the stakes are so high that it is prohibitively costly forher to make “the wrong” decision (x= �1). Because executive decision-making corresponds to thelimiting case when ↵

⇤ ! 1, my analysis of supervisory decision-making when ↵

i

↵

⇤ characterizesthe results under executive decision-making.

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p2(·) ! 1 as ri ! 0.7

Successful communication is more likely the more effort the expert exerts onpersuasion (p1(·) > 0) and the more attention the DM devotes to his message (p2(·) >0). In short, communication is a team effort. Because communication efforts arecomplements (p12(·) > 0), an expert’s return from expending effort is higher whenthe DM listens more attentively, and the DM’s return from paying attention is higherwhen the expert makes a greater effort to explain. The formulation encompassescommunication technologies with the property p (0, ri) 6= 0: Even if an expert makesno effort to transmit information to the DM, it is possible for her to find the relevantinformation by herself.8

Communication is costly to both parties. The DM’s attention is scarce,P

i ri 1, so the cost is attention substitution: Paying more attention to one action neces-sarily comes at the expense of others. An expert’s cost of persuasion effort is givenby c (si).

Assumption The function c (si) is twice continuously differentiable on (0, 1) andsatisfies c

0(·) > 0 and c

00(·) > 0, as well as the Inada conditions c

0(·) ! 0 as si ! 0,

and c

0(·) ! 1 as si ! 1.

The persuasion efforts and the attention allocation are chosen simultaneouslyand non-cooperatively. I refer to the above game as the deliberation stage.

I determine the Nash equilibrium for this game, and then analyze how the expertsaffect each other in equilibrium.

Lemma 1. If ↵1,↵2 ↵

⇤ , there is a unique equilibrium (r

⇤1, s

⇤1, s

⇤2), which is inte-

rior. If ↵1 ↵

⇤< ↵2, there is a unique equilibrium (r

⇤1, s

⇤1, 0). If ↵1,↵2 > ↵

⇤, thereis a unique equilibrium (r

⇤1, 0, 0).

An expert’s behavior hinges on whether the DM uses an opt-in rule or an opt-outrule for the decision he is vested in. When ↵i ↵

⇤, the DM uses an opt-in rule withrespect to Ai. Her default is not to take Ai, but she departs from this default—optsin—if she learns that xi = x. Hence, expert i has an incentive to communicate withher. Thus, when ↵1,↵2 ↵

⇤, each expert solves

max

si

{d↵ip (si, ri)� c (si)} ,

7The subscripts refer to the derivative of a function with respect to the ith argument.8Dewatripont and Tirole (2005) study the particular complementary technology p(s

i

, r

i

) = s

i

r

i

,and thus do not allow for the DM to find the relevant information by herself.

8

and the DM’s problem is

max

r12[0,1]{x (↵1p (s1, r1) + ↵2p (s2, 1� r1))} . (1)

In the unique equilibrium, both experts communicate, and the DM pays attention toboth. The Inada condition rules out corners; global concavity of p (si, ri) guaranteesuniqueness.

In contrast, when ↵i > ↵

⇤, the receiver uses an opt-out rule with respect toAi. Her default is to take Ai, but she departs from this default—opts out—if shelearns that xi =x. Because communication can only persuade the DM not to takeAi, expert i makes no effort. The DM can nevertheless devote attention to Ai; thatis, she can engage in (one-sided) information acquisition. If ↵1 ↵

⇤, ↵2 > ↵

⇤, herproblem is

max

r12[0,1]{x↵1p (s1, r1) + ↵2x+ (1� ↵2)x� p (s2, 1� r2) (1� ↵2)x} .

In the unique equilibrium, expert 1 exerts effort, expert 2 is passive, and the DMcommunicates with expert 1 about A1 and devotes some attention to acquiring infor-mation about A2. The distinction between one-sided and two-sided communicationarises endogenously.

Crowding out How well an expert fares in the deliberation stage not only dependson how attractive his own action seems to the DM, but also on the other expert’sattractiveness.

Proposition 1 (Crowding out). Fix expert 2’s attractiveness, ↵2. If expert 2 wantsthe DM’s attention (↵2 ↵

⇤), his expected utility is a strictly decreasing function ofthe attention given to expert 1, r⇤1 (↵1). If expert 2 does not want the DM’s attention(↵2 > ↵

⇤), his expected utility is a strictly increasing function of the attention givento expert 1, r⇤1 (↵1).

As r⇤2 (↵1) = 1�r

⇤1 (↵1), a change in ↵1 that causes the DM to pay more attention

to expert 1 in equilibrium crowds out attention to expert 2. When expert 2 wantsthe DM’s attention, this crowding out harms him; otherwise, it benefits him. Thus,the presence of expert 1 imposes a negative or positive externality on expert 2, andthe size of this externality is captured by r

⇤1 (↵1).

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Corollary 1. Fix expert 2’s attractiveness, ↵2. Expert 2’s expected utility is non-monotonic in the attractiveness of expert 1, ↵1.

This follows from the fact that the attention the DM devotes to expert 1 inequilibrium, r

⇤1 (↵1), is non-monotonic in ↵1: r

⇤1 (↵1) increases for ↵1 2 (0,↵

⇤),

falls at ↵

⇤, and decreases for ↵1 2 (↵

⇤, 1). If expert 2 wants the DM’s attention

(↵2 ↵

⇤), EUExp2 (↵1) is negatively related to r

⇤1 (↵j); otherwise, the reverse holds.

This is illustrated in Figure 1.

Figure 1: Crowding Out

Note: The DM’s attention devoted to Expert 1 in equilibrium, r

⇤1 (↵1), for a given ↵2, is non-

monotonic in Expert 1’s attractiveness (left panel). This makes Expert 2’s utility nonmonotonicin ↵1: When Expert 2 desires attention (↵2 ↵⇤), r

⇤1 (↵1) represents a negative externality on

Expert 2, so EU

Exp2 (↵1) is negatively related to r

⇤1 (↵1) (middle panel). When Expert 2 does not

desire attention (↵2 > ↵⇤), r⇤1 (↵1), represents a positive externality on Expert 2, so EU

Exp2 (↵1)is positively related to r

⇤1 (↵1) (right panel).

2.2 The selection stage

We now add a pre-play stage by introducing cue communication, as in Dewatripontand Tirole (2005).

Set-up The DM can pay attention to two distinct actions in the deliberationstage.9 When more than two experts seek the DM’s attention, selection becomes

9The assumption that the DM can only devote attention to t topics can, in this context, bethought of as a lower bound r on the amount of (nonzero) attention that the DM can devote toany one topic, r

i

2 {0} [ [r, 1] for all i. This limits the number of topics she can deliberate on tot ⌘ t (r) 2 N. This assumption is appealing in the presence of a large number of topics; in practice,it is not possible to devote only a split second to each of (infinitely) many sources. The choice of

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an important issue. To capture this, I add a pre-play stage in which the DM mustselect at most two experts for the deliberation stage. I refer to the pre-play stageas the “selection stage.”10

N = N↵ + N↵ experts can enter the competition to beselected. Of these, N↵ propose actions of high quality (↵ = ↵) and N↵ of low quality(↵ = ↵ < ↵). N↵ > t is finite; N↵ is infinite. I set parameters such that all expertswant attention, ↵ < ↵ < ↵

⇤.As in Dewatripont and Tirole (2005), each expert’s quality is his private informa-

tion. Hence, an expert may be willing to signal the quality of his action if it helps himget selected, and the DM may be willing to read such signals before choosing whichactions to deliberate on, that is, which experts to communicate with. Specifically, atcost qS > 0, an expert can send a cue (signal) that contains hard information aboutthe quality of his action. Upon receiving a cue, the DM decides whether to processit, at cost qR > 0, to learn the action’s quality. No expert can be selected withouthaving sent a cue; hence, qS can be thought of as an entry cost. As I explain below,this last assumption is not crucial.

I solve this game, deliberation stage plus selection stage, for perfect Bayesianequilibrium. Furthermore, I focus on the equilibria favored by the DM, that is,those with the maximum number of high-quality entrants.

Information overload As would be expected, when the cost of entry decreases,the supply of experts—and hence the number of actions the DM can choose among—increases. However, as the DM’s choice set grows, her expected utility first increases,but then decreases. Proposition 2 states this key result:

Proposition 2 (Information overload). As qS ! 0, the DM receives more cues buteventually processes less. Her expected utility first increases and then decreases.

It is instructive to describe how equilibrium behavior in the selection stage

t = 2 is merely one of convenience; I show in the proof of Proposition 2 that all results go throughfor any finite t.

10The distinction between the selection stage and the deliberation stage is founded in cognitivescience. As Cohen (2011, p.1) writes, the distinction between “attentive processing” (the deliber-ation stage) and “pre-attentive processing” (the selection stage) is logically inherent in the notionof selective attention: “A fundamental empirical phenomenon in human cognition is its limitation. . . One trademark of a limited system is its need for selection . . . Any type of selection pre-supposes the availability of some information in order to perform the very selection. Thus, some‘pre-attentive’ processing must be performed prior to the operation of selective attention, and itsoutput is used for the selection. The distinction between pre-attentive and attentive processing isessential in the study of selective attention.” In this model, the cues represent the information uponwhich pre-attentive processing is performed.

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changes as the cost of sending cues, qS , falls from prohibitively large to negligi-bly small. The impact on the DM’s expected utility is illustrated in Figure 2, wherea decrease in qS represents a movement from right to left on the x-axis.

• For high enough a qS , cues are so expensive that no expert enters.

• As qS falls, it at some point becomes attractive for some high-quality expertsto enter. Here, the cues in themselves are a signal of high quality, so the DMneed not process them but can select her communication partner(s) for thedeliberation stage from the pool of entrants at random. (This relies on theassumption that the DM can observe that a cue was sent even if she does notassimilate it. If we relax this, the economic insights remain valid, as I explainbelow.) In this signaling outcome, the DM’s welfare increases as qS falls so longas the number of entrants is smaller than two—or more generally, smaller thanthe number of experts she can communicate with in the deliberation stage;otherwise, the DM’s welfare remains constant. This is captured in Figure 2:As qS decreases, the DM benefits from an expansion in information supply solong as qS > qS . Then, as qS falls further (but remains above q

S), the DM has

access to (at least) 2 high-quality experts, but no low-quality experts, so herexpected utility remains flat as qS falls further.

• As qS falls below q

S, some low-quality experts find it attractive to enter as

well. A signaling equilibrium, in which a random pick from among the entrantsensures a high-quality expert for the deliberation stage, no longer exists. TheDM reacts in either of two ways: Either she continues to randomize and simplyaccepts the lower (average) expert quality or, if qR is not too high, she readscues with positive probability to screen out low-quality experts. So as not tomake the selection stage trivial, I focus on qR low enough for the DM to engagein active screening. Clearly, her welfare decreases as qS falls, as it becomesharder to spot high quality. Already, the arrival of more cues—essentially,access to more information—makes the DM worse off. The next stage is merelythe copestone.

• As qS vanishes, the avalanche of low-quality cues reduces the average quality inthe entrant pool so much that screening becomes futile—high quality becomesthe proverbial needle in the haystack. As a result, there is neither signalingnor screening, just pooling : the DM gives up on active selection and accepts

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that she is all but bound to meet low quality in the deliberation stage. Herexpected utility thus approaches U⇤

, her expected utility from communicatingwith 2 low-quality experts (on 2 low-quality actions) in the deliberation stage.

Thus, at a certain point, as the competition for the DM’s attention increases andshe gets more information, she processes less of it—or tunes out—and fares worse.I refer to this phenomenon—the more cues the DM gets, the less she processes, andthe worse she fares—as information overload.11

Figure 2: Information overload

A movement from right to left on the x-axis represents a decrease in the cost of entry, qS

. As q

S

decreases, the DM first benefits from an expansion in information supply (so long as q

S

> q

S

).Then, the DM has access to (at least) 2 high-quality experts, but no low-quality experts, so sheobtains her maximum possible (decision) payoff, U

⇤. As the cost of entry falls below q

S

, lowtypes join the battle for access to the DM’s attention, and the her expected utility falls below U

⇤.When q

S

! 0, the number of low types who enter tends to infinity, so the DM ceases to screenexperts. Her decision payoff falls, as she relies on lower quality of information on average. Thus,when information becomes cheap enough, the more information she gets, the less information sheprocesses, and the worse she fares.

It is instructive to make precise how (the idea of) information overload is relatedto (the idea of) limited attention. To this end, consider this quote by Simon (1971):

What information consumes is rather obvious: It consumes the at-tention of its recipients. Hence a wealth of information creates a povertyof attention, and a need to allocate that attention efficiently among theoverabundance of information sources that might consume it. [p. 40-41]

11The signaling and the screening outcomes arise independent of our assumption that there areinfinitely many experts of low quality. The pooling outcome requires that N

↵

, the number oflow-quality experts, is sufficiently large relative to N

↵

, the number of high-quality experts.

13

Thus, attention, in limited supply, becomes a scarcer resource in relative terms whenconfronted with more information. But this does not imply information overloador that more information provided can decrease knowledge attained. The idea ofinformation overload is that a wealth of information not only “creates. . . a need toallocate that attention” (emphasis added) but actually impairs the ability to do soefficiently.

Alternative assumptions If I instead assume that the DM cannot observethat a cue was sent unless she incurs a cost to read it, the economic insights remain:She must open exactly two cues so long as only high-quality experts enter; then, shemust either open exactly two cues but rely on information of lower quality, or openmore than two cues on average to identify two high-quality experts. In either case,her expected utility remains constant when only high types enter and decreases withthe number of low types.

Further, the equilibria described above exist even if we relax the assumptionthat an expert must send a cue to enter. However, in that case, there is a furtherequilibrium for qS ! 0 in which the experts cease to send cues, aware that they areno longer processed, and the DM picks randomly from the entire pool of experts.Still, the DM favors the equilibrium in which she picks randomly from a subset ofexperts—which includes all high-quality experts—that send a cue, because it offersbetter odds of picking a high-quality expert.

Also, we need not assume differences in quality. Instead, suppose experts investin quality. Specifically, suppose all N experts begin with low quality (↵ = ↵) butcan invest in high quality (↵ = ↵) at some cost c > 0 before entering. Proposition 2implies that information overload frustrates investment in high quality. Intuitively,the value of quality is reflected in the expected utility difference between a highand a low type. For prohibitive qS , both types expect to earn zero, so there is noincentive to invest in quality. As qS falls, if c is not too high, some invest in qualityand send cues. But as qS ! 0, information overload erodes the premium on quality,so again no one invests. The supply of high quality collapses when it becomes toocheap to approach the DM. This is not because the DM ceases to value quality. Onthe contrary, she would like to treat high-quality experts preferentially; however, shein unable to do so when finding them amounts to looking for a needle in a haystack.

14

Externalities as the driving force of information overload The immediatecause of information overload is that the quality of the proposed actions decreaseswith the quantity; actions worthy of deliberation become the proverbial needle in ahaystack. The deeper cause, though, is negative externalities: Entry is individuallyrational for each expert, even as it complicates the selection problem for the DM andspoils overall communication. This is because each expert ignores how his own entryaffects the communication environment as a whole. If the expert were identical forall actions, he would only send cues for two high-quality actions. In the decentralizedsetting, however, sending cues remains individually rational even as each cue sentaggravates the complexity of the DM’s selection problem up to a point where activeselection breaks down. This, in turn, frustrates the incentives to produce quality.Intuitively, it is as if the low-quality experts, each seeking to be noticed, pollute theDM’s attention field. Indeed, information overload is similar to pollution or con-gestion, and like them, may be amenable to efficiency-improving intervention. Thenext subsection turns to practical expressions of information overload, and discussesthe fact that a DM with limited attention may want to limit access to her attentionspace, even if that reduces her choice set. In a nutshell, she faces a trade-off betweencomprehensiveness and comprehensibility.

2.3 Information overload in practice

With recent advances in information technology, individuals face massive data viamore channels (phone, Internet, email, instant messages, etc.) and on more platforms(Facebook, Twitter, blogs, etc.). In the presence of information overload, such anabundance of information can be counterproductive. Indeed, a business research firmrecently nominated information overload as the “problem of the year,” and claimedthat it caused “a $650 Billion drag on the economy” by way of “lost productivity andinnovation.” The main concern is that an escalating quantity of information comeswith a declining average quality and that this inverse relationship between amountand relevance makes it harder to find “good” information.12 This makes selection, asin my model, a daunting issue, and suggets that consumers should value productsthat help reduce their choice sets and improve selection.

In this vein, several technology firms, including Microsoft, Intel, Google, andIBM, recently formed a nonprofit organization, the Information Overload Research

12The quotes in this paragraph are from Lohr (2007).

15

Group, to develop solutions to information overload.13 Google’s success formula, itsranking algorithm, implements pre-selection. And its ubiquity in the Internet, asgateway and gatekeeper, betrays the import of information overload. The logic ofpre-selection also underlies solutions such as email filters, ranking inbox messages byimputed importance, compiling communication histories for every sender, displayingemail portions to allow for fast screening, and sophisticated filing and search func-tions. Such ranking of electronic messages minimizes information overload; that is,it reduces the “economic loss associated with the examination of a number of non-or less-relevant messages” and distinguishes “communications that are probably ofinterest from those that probably aren’t” (Losee, 1998).

Information overload is not just a matter of Internet and emails. In a seminalstudy, Jacoby et al. (1974) explore how the quality of consumption decisions dependson “information load,” measured as number of brands as well as amount of informa-tion per brand provided. Their experiment shows that the ability to pick the bestproduct dropped off at high levels of information load. In Jacoby et al. (1973), acompanion paper, they further show that the subjects spent less time on processinginformation—or in their words, tuned out—once the information load exceeded acertain threshold.14 Many other experiments in organization science, accounting,marketing, and information science corroborate the notion that more informationcan impair cognitive processes and decisions (Edmunds and Morris, 2000; Epplerand Mengis, 2004).

Because information overload is a driving force behind innovations in commu-nication and information management, it is connected to recent research on choicearchitecture, that is, how the presentation of choices affects decisions (Thaler andSunstein, 2008). Cronqvist and Thaler (2004), for example, study the introductionof a new retirement savings plan in Sweden in 1993. Eligible Swedes were encouragedto choose five out of 456 funds, to which their savings would be allocated. One thirdof all eligibles did not make any active choice; their savings were instead allocated toa default fund (essentially a pre-selection by the government). Information overloadseems a likely reason for so many Swedes to rely on the default choice: Compar-ing hundreds of funds is a Herculean task for ordinary households, and one mightexpect many of them to resort to the default or make superficial active decisions.

13The Information Overload Research Group’s web site is http://iorgforum.org/.14In the same vein, Iyengar (2011) provides direct empirical evidence that a reduction in choices

can benefit decision-makers.

16

Indeed, studying the same Swedish reform, Karlsson et al. (2006) show that fundsthat (for exogenous reasons) were better represented in the fund catalogue—that is,have better “menu exposure”—received more active contributions.15

All of the above examples suggest that decision-makers can benefit from receiv-ing less information, despite the associated decrease in choice set. Indeed, in thepresence of information overload, there is a trade-off between variety and simplic-ity, or between comprehensiveness and comprehensibility. This evidence contrastswith models of decision-making under unlimited attention, where a larger choice setcannot make an individual worse off.

3 A Single Expert and Strategic Attention Manipulation

In the previous section, crowding out and information overload result from the strate-gic interaction between multiple experts, each of whom vouch for the DM to takeone distinct action. By making minor modifications to the original setting, thissection reinterprets the framework to study a DM who communicates with a singleexpert on one action that has several aspects. In the deliberation stage, the DM’smulti-tasking problem now stems from the fact that the DM must decide how toallocate her scarce attention between the various aspects of the action. In the se-lection stage, recall that in the “multiple experts”-setting analyzed in Section 2, theDM is unsure of each action’s ex ante appeal. The analogue in Section 3 is that theDM is unsure which aspects of the (single) action are financially relevant, and henceworth devoting attention to in the deliberation stage. Next, we introduce the set-upin detail.

3.1 Modified set-up: Deliberation and selection stages

The DM communicates with a single expert about one action, A, which has manydifferent aspects. Specifically, suppose the DM’s payoff from A can be expressed asthe sum of NR components: x =

PNR

1 xi. Each component takes the value x > 0

15In the same vein, studying a retirement savings plan in the United States, Beshears et al.(2013) show that making the decision problem less complex—by collapsing a multidimensionalproblem into a binary choice—increases enrollment in the plan. In a similar study, Choi et al.(2012) report that sending short email cues that draw attention to selective details of the savingsprogram significantly affects participation. Further, in the context of Denmark, Chetty et al. (2014)show that wealth accumulation is highly responsive to automatic retirement contributions, whicheffectively eliminate the need to process information ahead of making retirement contributions.

17

with probability ↵i and otherwise the value x < 0. So the expected payoff from A isE (x) = NR [↵x+ (1� ↵)x], where ↵ =

1N

R

PNR

1 ↵i. Similar to before, absent moreinformation, the DM takes A in the deliberation stage if and only if ↵ > ↵

⇤ ⌘ �xx�x .

Or put differently, if the DM can obtain more information, she uses an opt-in rule if↵ ↵

⇤ and an opt-out rule if ↵ > ↵

⇤.In addition to the NR components of the action A that are relevant to the action’s

payoff, there exist an additional N; components of A that are irrelevant to the DM’spayoff. I assume that the DM does not know which components are relevant16, andis potentially unaware of components per se. Intuitively, this captures a plausiblesituation: Inclined towards a particular choice, the DM might yet discover (finan-cially relevant) aspects that change her opinion. Thus, the DM is faced with two setsof questions: What components exist, and which ones are relevant (selection stage)?And how much attention should a given component receive (deliberation stage)?

As before, the expert’s communication incentives hinge on the DM’s decisionrule:

If the DM follows an opt-in rule, the expert must persuade her to take the action.To maximize the chances that the DM revises her beliefs upwards, and so opts in,the expert seeks to draw her attention to those aspects that are most likely to yieldfavorable information. That is, he sends her cues about, and exerts persuasion efforton, the topics with the highest ↵i.

By contrast, if the DM follows an opt-out rule, her default is to take action A,but she may depart from this default—and opt out—if she learns about any relevantaspects of the action that are unfavorable. Then, the expert wants to withholdthe relevant information. In the model, this means that, on his own accord, hewould never send a cue about any relevant aspect in the selection stage, since thiscould only induce the DM to devote attention to it in the deliberation stage, andsubsequently to change her mind and abandon the action after all. In this situation,a mandatory disclosure law would make a difference to the expert’s communicationstrategy: By mandating that the expert (firm) reveals any relevant financial aspectsof the product, it would be illegal to withold this information.

To formalize what happens when the expert is subjected to such a mandate,assume for simplicity that there is one relevant topic, NR = 1, that the number

16This is analogous to the assumption, in the setting with multiple actions presented in Section2, that the DM does not know which actions are of a high quality and which actions are of a lowquality.

18

of irrelevant topics N; is infinite, and that the DM can at most deliberate on twotopics (as in Section 2). These assumptions are merely simplifying; indeed, the below“strategic information overload”-result holds as long as the DM can devote attentiononly to a limited number of topics in the deliberation stage and N; is sufficientlylarge relative to NR. Finally, as before, it is costly for the DM to process cues:qR > 0. For convenience, I refer to cues about relevant (irrelevant) topics as relevant(irrelevant) cues.

3.2 Strategic information overload and the limitations of manda-

tory disclosure laws

When ↵ > ↵

⇤, the DM follows an opt-out rule. Thus, if the DM lacks access to—or isunaware of—the relevant topic, the expert has no reason to bring it to her attentionby sending a relevant cue. In fact, the expert is best off sending no cues at all. Nowsuppose that, by a disclosure mandate, the relevant cue must be sent.

Proposition 3 (Strategic information overload). Let ↵ > ↵

⇤. Suppose that disclo-sure laws mandate that the relevant cue is sent. As qS ! 0, the expert sends anincreasing swarm of irrelevant cues, and the DM’s expected utility decreases.

Proposition 3 is closely related to the information overload result in Section2; yet, the underlying economic mechanism is different. In Section 2, informationoverload resulted from various experts’ competition for attention. Proposition 3shows that a single expert, who faces no competition for the DM’s attention but whocan communicate with the DM about multiple aspects of the action he advocatesfor, strategically induces information overload in the DM when he is subjected toa disclosure provision that mandates drawing the DM’s attention to the financiallyrelevant aspect of the action (the product he vouches for the DM to buy).

Intuitively, the expert is afraid that the DM, by paying attention to the relevanttopic, might discover unfavorable information about the action and opt out. Toreduce the odds that the DM identifies — that is, selects — the relevant topic, thedisclosure mandate thus gives the expert an incentive to supply irrelevant aspects inthe selection stage, by sending out irrelevant cues, even though this is costly for him.A swarm of mostly irrelevant cues, in turn, thwarts the DM’s chance, and hence herincentives, to pinpoint the relevant topic. In other words, the expert intentionallyinduces information overload, which effectively conceals the inconvenient aspect thatthe firm was mandated to disclose in the first place. In a nutshell, the disclosure

19

mandate, which is intended to raise the DM’s awareness of the financially relevantaspect of the product that she considers, generates a response on the part of theexpert that in practice renders the DM financially illiterate, or obfuscates her.

As a result, the disclosure mandate has no impact on the DM’s final decision;she buys the product even if an understanding of the financially relevant aspectwould have pushed her to opt out, since strategic information overload makes heras uninformed in the presence of a mandate as she is in its absence. This sharplyillustrates that, when consumers’ attention is limited, simple disclosure rules can becompletely impotent and have no impact on social welfare.

Strategic information overload in practice USA Today recently ran an inter-nal study on the costs of maintaining a basic checking account at the ten largest USbanks and credit unions. While the most basic fees were found to be disclosed onthe institutions’ websites, many others were listed only in the “Schedule of Fees andCharges.” That, however, turned out to be difficult to find.

But even the world’s largest search engine couldn’t unearth a feeschedule for HSBC, TD Bank, Citibank and Capital One. To get theirfee information, we had to e-mail or call the banks.

Determined customers can search for information about fees in banks’official disclosure documents, but they’ll need a lot of time and a coupleof cups of coffee, too. An analysis of checking accounts for the 10 largestbanks by the Pew Health Group found that the median length of theirdisclosure statements was 111 pages. None of the banks provided keyinformation about fees on a single page...17

Note that the issue was not only that the “inconvenient” information was at timesunavailable. It was also that, even when provided, it was made available in a waythat made it costly to locate the relevant information; that is, in a way that inducedstrategic information overload in the customer. Ordinary customers would be hard-pressed to know not only where to look for relevant items but also what items tolook for. Similar conditions prevail in other countries. In 2008, the website ThisIs Money cited a warning by the British consumer and competition authority, theOffice of Fair Trading, that

17Tilghman, Molly, and Sandra Block. 2011. “Finding Info on Bank Fees May Take Digging.” USAToday, October 21, http://www.usatoday.com/money/perfi/credit/story/2011-10-20/comparing-bank-fees/50845842/1

20

[credit card] providers can add to the problem knowing that con-sumers cannot process complex information . . . They can create “noise”by increasing the quantity and complexity of information, which makesit difficult for consumers to see the real price.18

Note that the concern here is the complexity of information; even if a proudct itselfis simple, information overload effectively presents it in an overly complex fashion,by hiding it in very long disclosure statements. The financial products market seemsrife with such practices. Credit cards are perhaps the most widely debated example.As quoted in a 2009 Reuters article, President Obama said “No more fine print, nomore confusing terms and conditions,” in a meeting with US credit card companyexecutives on consumer protection regulation.19

These examples underscore an interesting aspect that elude previous commu-nication models: mandatory disclosure is not a panacea. In the above examples,the communication problem is neither a willful misrepresentation (“cheap talk”) northe withholding of facts (“strategic non-disclosure”). Here, the banks are mandatedto provide fee information; cheap talk or non-disclosure are illegal and would haveserious consequences ex post. Still, this does not mean that consumers become well-informed. Even when information is hard and disclosed, senders can still—throughstrategic attention manipulation—conceal what is relevant by manipulating the sheeramount of information. Thus, simple disclosure rules like the Truth in Lending Actmay prove completely ineffective when consumers’ attention is limited.20

3.3 Complexification and the limitations of “simple labels” for con-

sumer protection

The previous subsection illustrates that too much disclosure can be as much ofa concern as too little; a surfeit of details can prove as uninformative as a dearththereof. But if disclosure mandates can backfire because firms can induce informationoverload, intuition suggests an immediate solution: to mandate not only what firms

18Daily Mail Reporter and Sean Poulter. 2008. “Credit Card £400m Small Print Rip-Off,” ThisIs Money, October 29, http://www.thisismoney.co.uk/money/cardsloans/article-1619869/Credit-card-400m-small-print-rip-off.html

19Alexander, David, and John Poirier. 2009. “Obama Calls for Credit Card Re-forms,” Reuters, April 23, http://www.reuters.com/article/2009/04/23/us-obama-creditcards-idUSTRE53M10720090423

20The Truth in Lending Act, of 1968, is a federal law intended to ”safeguard the consumer inconnection with the utilization of credit by requiring full disclosure of the terms and conditions offinance charges in credit transactions or in offers to extend credit.”

21

disclose, but also that it is disclosed in an easy-to-understand way. Indeed, morerecent disclosure rules often are of this flavor: Credit card companies must nowdisclose key details of the products they offer in a salient fashion, health insurancecompanies must provide a Summary of Benefits and Coverage (SBC) of each planoffered, and food producers must adhere to standardized labeling regulations todeclare all ingredients and the caloric content of each serving, for example.

In this subsection, I use the theoretical framework to analyze the strategic re-sponse on the part of an expert (firm) when faced with a mandate to provide easy-to-understand information about the product it sells. As we will see, even detailedmandates to disclose easy-to-understand information can backfire, by inducing a“complexification” of the underlying product itself.

I show this by extending the model with a single expert to allow the DM’s pay-off from the (single) action to be comprised of many components, and to allow theexpert to manipulate that composition so long as the total payoff stays constant.Intuitively, such payoff-equivalent variations amount to changing the number of fi-nancially relevant aspects of the product. Below, I formalize this:

To allow for complexification, I introduce an option for the expert to design A

in a way that makes its payoff less transparent. As before, the DM’s payoff is—atleast initially—the sum of NR components, x =

PNR

1 xi, and each component takesthe value x > 0 with probability ↵i and otherwise the value x < 0. However, theexpert can now recompose the payoff structure into any form y =

PN1 yj so long as

the total (realized) payoff is invariant: y = x. I call y a payoff-equivalent variation(of x). Crucially, I assume that the DM does not know the “original” composition,{xi}.21 In any case, absent communication, such variation does not matter; it affectsneither the DM’s decision nor her welfare.

As an illustration, consider two simple mathematical operations the expert canuse to create payoff-equivalent variations. One is to divide each xi into m parts,Pm

h=1 xih = xi, so that the new payoff structure has N = mNR components: y =

PNR

i=1

Pmh=1 xih =

PNj=1 yj . For example, instead of incorporating total expenses into

one salient item, such as the monthly rent, a landlord might disaggregate them intovarious fees, such as for maintenance, utilities, move-in or move-out, parking, laundry

21Alternatively, I could assume that she does not know how a given payoff-relevant variation isrelated to the original composition. Either assumption captures situations of the following kind: Aconsumer is inclined to buy a good based on superficial information. That said, she is not aware ofall aspects, such as hidden costs, that could influence her decision. Further, discovering one aspectis not necessarily informative about undiscovered ones.

22

room, or other administrative services. The other operation is to add componentsthat neutralize each other, as in y =

PNR

i=1 xi +PN

R

i=1 xi �PN

R

i=1 xi =PN

j=1 yj whereN = 3NR. A real-world example is a purchase involving nominal price, fees, taxes,discounts, bonuses, rewards, and so forth, which partly offset each other. In bothexamples, seeing one component is not informative about others.

Even if the total expected payoff remains unchanged, the payoff compositionmatters for communication. As before, suppose the DM cannot deliberate on morethan a certain number of topics (in keeping with the previous framework, say, two).A proliferation of components then causes more relevant information to slip herattention. Furthermore, disaggregating the payoff can make each component, in andof itself, less important. To see this point, let xih =

xi

m for all i in the first (landlord)example above. Increasing m leaves the total payoff, y, unchanged but shrinks everycomponent, x

i

m . Crucially, this reduces how much the DM can learn from a givennumber of components.

Payoff-equivalent variation is thus a means to manipulate the DM’s learningprocess. How the expert uses such means hinges, like before, on the DM’s decisionrule. If she uses an opt-in rule, the expert wants to help her learn more about A.He would set N 2—such that no relevant aspect escapes deliberation—and exertcommunication effort on all the components. That is, he would simplify the payoffstructure and strive to explain.

By contrast, if the DM follows an opt-out rule, the expert wants to do the exactopposite. He would increase the number of components, even if it were costly to doso, only to thwart learning. Suppose he must pay v > 0 to raise N by one.

Proposition 4 (Complexification). Let ↵ > ↵

⇤. Suppose all relevant cues are sentout. As v ! 0, the expert sets N ! +1, and the DM’s expected utility falls toE (x).

As v ! 0, the expert increases N such that more relevant information escapesthe DM’s attention, given that she can only deliberate on a limited number of com-ponents. At the same time, he disaggregates the payoff to reduce the amount ofinformation she can possibly wrest from any given component. In the limit, evenwhat she can learn from the components that she is capable of studying becomes sotrivial that it no longer affects her decision: She chooses what she would have cho-sen without the information. Intuitively, the expert makes the action unnecessarilycomplex, and thereby succesfully prevents the DM from getting the full picture.

23

Complexification thus has same welfare consequences as inducing informationoverload — it can eradicate all intended welfare gains from mandated informationprovision, even if the mandate prohibits unclear communication.

Complexification in practice According to Edward L. Yingling, president andchief executive of the American Bankers Association (ABA), Obama recently urgedthe companies “to issue a simple credit card product” (emphasis added). A yearearlier, after similar comments by Federal Reserve Chairman Ben S. Bernanke “thatimproved disclosures alone cannot solve all of the problems consumers face in tryingto manage their credit card accounts,” the ABA and other industry representativeshad signaled strong opposition to such interventions.22

The UK financial regulator, the Financial Services Authority (FSA), is alsoquoted saying

[P]roviders of financial products may gain from the lack of price trans-parency about their products. [. . . ] It may be in the provider’s interestto increase the complexity of the product charges.

Other examples of complexification come from the food industry, where legislationmandates that all ingredients be listed in a food declaration; however, “incidentaladditives” need not be listed. This has provided food producers with incentivesfor complexification, most prominently by removing an ingredient that consumersmay be want to avoid, and replacing it with one or several incidental additives thataccomplish the same effect in food but that remain invisible (provided each of themis included in small enough a quantity). A recent example is Starbuck’s use of a coloradditive derived from the animal world classified as an incidental additive instead ofan ingredient, which effectively caused a non-vegan product to be labeled vegan andcaused a stir among vegan consumers.23 Similarly, widespread consumer awaranessof the dangers of BPA has spurred development of a new range of plastic productslabelled “BPA Free”; however, these products contain BPA replacement substancesthat have produced similar health problems as BPA itself.24 Clearly, in the absence

22Labaton, Stephen. 2008. “U.S. Seeks New Curbs on Credit-Card Practices,” New York Times,May 3, http://www.nytimes.com/2008/05/03/business/03credit.html

23Source: Fortney D. Beware: Starbucks’ Soy Strawberries & Creme Frappuccino Is NOT vegan[weblog entry]. This Dish Is Veg (14 Mar 2012). Available: http://goo.gl/kPpj5 [accessed 11 Mar2016].

24Source: LaMotte S. BPA-free plastic alternatives may not be safe as you think (1 Feb2016). Available: http://www.cnn.com/2016/02/01/health/bpa-free-alternatives-may-not-be-safe/[accessed 11 Mar 2016].

24

of mandated disclosure of the ingredients in food, none of these complexificationshad been necessary; but in the presence of this disclosure rule, they representedstrategic responses on the part of food producers.

As opposed to the examples in the previous subsection, which showed manipula-tion of the complexity of information, the examples provided here refer to manipu-lation of the complexity of the product itself. Both when it comes to the complexityof information and the complexity of the product itself, the danger is that misguideddecisions affect consumers’ risk of getting into debt or of buying food products thatthey wish to avoid.

However, while complexity of information has a simple legislative recipie – man-date firms to provide easy-to-understand information – complexification is muchharder to address from a legislative perspective. In fact, Proposition 4, along withthe real-world examples of complexification in this subsection, highlight that eventhe most elaborate disclosure rule – which specifies exactly how easy-to-understand-information should be provided – may be completely ineffective for the consumer.This is because any regulation at the communication level proves futile if the sellercan modify the object of communication in a way that makes it intellectually chal-lenging to grasp, even with all details correctly disclosed. While the communicationis correct, the matter to be decided becomes too complicated. In such a case, to beeffective, regulation may have to target the object of communication per se, that is,product design. This, as the examples illustrate, is a much thornier issue.

4 Conclusion

Consumers’ attention limitations give firms incentives to manipulate prospectivebuyers’ allocation of attention. To the best of my knowledge, this paper is the first tomodel such attention manipulation. In its presence, competitive information supplycan reduce consumer knowledge by causing information overload. Moreover, a singlefirm subjected to a disclosure mandate may deliberately induce information overloadto obfuscate financially relevant information, or engage in product complexificationto bound consumer financial literacy.

These findings demonstrate that attention limitations matter crucially for whetherdisclosure regulation improves consumer welfare: Disclosure rules that would bewelfare improving for agents without attention limitations can prove impotent forconsumers with limited attention. Obfuscation suggests a role for rules that man-

25

date not only the content but also the format of disclosure; however, even rulesthat mandate disclosure of “easy-to-understand” information are ineffective againstcomplexification, which may call for regulation of product design.

An interesting avenue, not pursued in the present paper, is that heterogeneityin attention constraints may provoke different forms or degrees of attention ma-nipulation. Banerjee and Mullainathan (2008) posit that the poor are subject totighter attention constraints than the rich, who can afford better technologies tofree up attention. They then show that this induces differences in productivity thatamplify the differences in initial endowment; inequality breeds more inequality. Sim-ilarly, Allcott and Taubinsky (2015) and Taubinsky and Rees-Jones (2015) documentconsiderable heterogeneity in attention limitations, and Taubinsky and Rees-Jones(2015) show that lower-income individuals have stronger attention limitations. Thecurrent paper’s findings suggest that the problem may be even worse: The poormay not only start out with tighter attention constraints, but may also find theirlimited attention exploited, more so than the rich. In short, the tighter constraintsmay make them less productive and more manipulable. Manipulation is perhapsthe more worrisome problem in that it is, as shown in this paper, prone to createexternalities, and thus constrained inefficient outcomes. But such questions are leftfor future research.

26

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30

A Proofs

A.1 Proof of Lemma 1

Claim A1.1 When ↵1,↵2 ↵

⇤ , there exists a unique, interior equilibrium(r

⇤1, s

⇤1, s

⇤2) 2 (0, 1)

3 of this game.

Proof of Claim A1.1 When ↵1,↵2 ↵

⇤, the problem of Si, i 2 {1, 2} is givenby

max

si

�0{d↵ip (si, ri)� c (si)} .

For a given (conjectured) br1, Si’s first-order condition is given by

d↵ip1 (si, bri) = c

0(si) . (2)

Because c

0(si) ! 0 as si ! 0, and because c

0(si) ! 1 as si ! 1, the range of the

right-hand side (RHS) is (0,1). As d > 0 and p1 (si, bri) > 0 for all bri 2 [0, 1], theleft-hand side (LHS) is strictly positive for ↵i 2 (0, 1] . Since p (si, bri) is concave insi and c (si) is convex, LHS is decreasing in si and RHS is increasing in si. Theseobservations yield that there exists a unique solution s

⇤i (bri) 2 (0, 1) to this equation.

Clearly, the best reply function s

⇤i (bri) is monotonically increasing if p12 > 0, and

monotonically decreasing if p12 < 0. Because the function p(·) is concave and c(·) isconvex, this solution is the solution to the maximization problem (the second-ordercondition holds).

The problem of the DM is given by

max

{r1,r2}2[0,1]2x (↵1p (s1, r1) + ↵2p (s2, r2)) s.t. r1 + r2 = 1

, max

r12[0,1]x (↵1p (s1, r1) + ↵2p (s2, 1� r1)) .

For given (conjectured) bs1 and bs2, her first-order conditions (FOCs) are given by

x (↵1p2 (bs1, r1)� ↵2p2 (bs2, 1� r1)) = 0, (3)

where x > 0.We substitute the experts best reply functions from (2) into (3) and obtain

↵1p2 (s⇤1 (r1) , r1) = ↵2p2 (s

⇤2 (1� r1) , 1� r1) . (4)

31

An equilibrium which is interior must satisfy (4). We will now discuss existence anduniqueness of equilibria in this communication game.

First, we show that p2 (s⇤1 (r1) , r1) is monotonically decreasing in r1, i.e., that itsderivative is negative. Differentiating p2 (s

⇤1 (r1) , r1) with respect to r1 yields

p21 (s⇤1 (r1) , r1) s

⇤01 (r1) + p22 (s

⇤1 (r1) , r1) . (5)

Differentiating sender 1’s equilibrium condition, (2), with respect to r1 yields

d↵i

�p11 (s

⇤1 (r1)) s

⇤01 (r1) + p12 (s

⇤1 (r1))

�= c

00(s

⇤1 (r1)) s

⇤01 (r1) ,

s

⇤01 (r1) =

d↵ip12 (s⇤1 (r1))

c

00(s

⇤1 (r1))� d↵ip11 (s

⇤1 (r1))

(6)

Inserting (6) into (5) yields that the derivative of p2 (s⇤1 (r1) , r1) is negative if andonly if

d↵ip12 (s⇤1 (r1)) p12 (s

⇤1 (r1))

c

00(s

⇤1 (r1))� d↵ip11 (s

⇤1 (r1))

+ p22 (s⇤1 (r1) , r1) < 0.

Using the fact that (c

00(s

⇤1 (r1))� d↵ip11 (s

⇤1 (r1))) is strictly positive, we rearrange

to obtain

d↵ip12 (s⇤1 (r1)) p12 (s

⇤1 (r1)) < �p22 (s

⇤1 (r1) , r1)

�c

00(s

⇤1 (r1))� d↵ip11 (s

⇤1 (r1))

�

p12 (s⇤1 (r1)) p12 (s

⇤1 (r1)) < p11 (s

⇤1 (r1) , r1) p22 (s

⇤1 (r1) , r1)�

1

d↵ip22 (s

⇤1 (r1) , r1) c

00(s

⇤1 (r1)) .

Because � 1d↵

i

p22 (s⇤1 (r1) , r1) c

00(s

⇤1 (r1)) > 0, this condition is implied by global con-

cavity. This establishes that p2 (s⇤1 (r1) , r1) is monotonically decreasing in r1.

Second, we show that there exists a unique interior equilibrium. Defining g(r1) ⌘p2 (s

⇤1 (r1) , r1) and h(r1) ⌘ p2 (s

⇤2 (1� r1) , 1� r1) we rewrite (4) as

↵1g (r1) = ↵2h (1� r1) . (7)

Step 1 of this proof established that g (r1) is decreasing. An analogous argumentestablishes that h (r2) = h (1� r1) is decreasing in r2 = 1 � r1 (increasing in r1).

32

Further, because g (r1) = p2 (s⇤1 (r1) , r1), the Indada condition

for all si 2 [0, 1] : p2 (si, ri) > 0 for all ri 2 [0, 1) and p2 (si, ri) ! 0 as ri ! 1 (8)

yields that

g (r1) > 0 for all r1 2 [0, 1) and g (r1) ! 0 as r1 ! 1 (9)

h (r2) > 0 for all r2 2 [0, 1) and h (r2) ! 0 as r2 ! 1,

where the latter can be re-written

h

0(1� r1) > 0 for all r1 2 (0, 1] and h

0(1� r1) ! 0 as r1 ! 0, (10)

By (9), when r1 tends to one, LHS of (7) tends to zero and RHS is strictly greaterthan zero. By (10), when r1 tends to zero, RHS tends to zero and LHS is strictlygreater than zero. Thus, there exists an interior equilibrium r

⇤1 2 (0, 1), as these

must cross. Moreover, they cross at most once, so the solution r

⇤1 is unique. By

arguments analogous to those above, second-order condition is satisfied.From the above two steps, we have that the unique interior equilibrium is given

by (r

⇤1, s

⇤1, s

⇤2) 2 (0, 1)

3, where r

⇤1 2 (0, 1) is the solution derived above, s⇤1 = s

⇤1 (r

⇤1),

and s

⇤2 = s

⇤2 (1� r

⇤1).

Third, we show that there exists no equilibrium in which the DM devotes allher attention to only one of the experts. Suppose that there exists some equilibriumin which r

⇤i = 0 for some Si, w.l.o.g. for S1. However, (8) yields that, for any s1,

@p(s1,r1)@r1

! 0 as r1 ! 1 and @p(s2,r2)@r2

> 0 as r1 ! 1. Thus, r⇤i = 0 cannot be optimalfor the DM . Note that this is the case even if s⇤1 = 1.

Claim A1.2 When ↵2 ↵

⇤< ↵1, there exists a unique equilibrium (r

⇤2, s

⇤2, 0).

Proof of Claim A1.2 When ↵1 > ↵

⇤ and ↵2 ↵

⇤, S1’s problem is given by

max

s1�0{d� p (s1, r1) (1� ↵1) d� c (s1)} .

Because the first-order derivative w.r.t. s1 is negative, s⇤1 = 0.The problem of S2 is identical to the experts’ problem in the case when ↵1,↵2

↵

⇤ above. Thus, S2’s (unique) best reply function s

⇤2 (br2) is monotonically increasing

33

if p12 > 0 and monotonically decreasing if p12 < 0.The problem of the DM is given by

max

r12[0,1]{↵1x+ (1� ↵1)x� p (s1, r1) (1� ↵1)x+ x↵2p (s2, 1� r1)} .

For given (conjectured) bs1 and bs2, her first-order conditions (FOCs) are given by

, �x (1� ↵1) p2 (bs1, r1) = x↵2p2 (bs2, 1� r1) .

We substitute in the experts’ best reply functions and obtain

� x (1� ↵1) p2 (0, r1) = x↵2p2 (s⇤2 (1� r1) , 1� r1) . (11)

Because �x (1� ↵1) > 0 and p2 (0, r1) is decreasing in r1, LHS of (11) is decreasingin r1. Replicating the steps in the proof of case 1 above establishes that RHS isincreasing in r1. As the Inada conditions stated in the proof of case 1 are defined forall s1 2 [0, 1], and hence for s1 = 0, an analogous argument yields that there exists aunique solution r

⇤1 2 (0, 1) to (11). Thus, there exists a unique interior equilibrium

(r

⇤2, s

⇤2, s

⇤1) 2 (0, 1)

2 [ {0} of this game. Moreover, replicating the steps in the proofof case 1 establishes that there exist no equilibrium in which r

⇤i = 0 for some i.

We note that in this equilibrium, the DM engages in (one-sided) informationacquisition relating to A1, i.e. she devotes some attention to this project even thoughS1 does not make any communication effort. In contrast, DM and S2 engage in two-sided communication.

Claim A1.3 When ↵1,↵2 > ↵

⇤ for, there exists a unique equilibrium (r

⇤1, 0, 0) 2

(0, 1) [ {0} [ {0} of this game.

Proof of Claim A1.3 Both experts problems are given by the problem of S1 inthe proof of Claim 2. Hence, s⇤1 = s

⇤1 = 0. The problem of DM is given by

max

r12[0,1]{↵1x+ (1� ↵1)x� p (s1, r1) (1� ↵1)x+ ↵2x+ (1� ↵2)x� p (s2, 1� r1) (1� ↵2)x} .

For given (conjectured) bs1 and bs2, her first-order conditions (FOCs) are given by

, �x (1� ↵1) p2 (bs1, r1) = �x (1� ↵2) p2 (bs2, 1� r1) .

34

We substitute in the experts’ best reply functions and obtain

(1� ↵1) p2 (0, r1) = (1� ↵2) p2 (0, 1� r1) . (12)

An analogous argument to those in the proofs of Claim A1.1 and Claim A1.2 yieldsthat there exists a unique solution r

⇤1 2 (0, 1) to (12).

A.2 Proof of Proposition 1

Claim A2.1 Fix the attractiveness of expert 2’s action, ↵2. The DM ’s attentiondevoted to Expert 1, r⇤1 (↵1), is non-monotonic in ↵1.

Proof of Claim A2.1 When ↵1 ↵

⇤, an increase in ↵1 affects the DM (andExpert 1) in two ways. First, it becomes more likely that the DM benefits fromA1. This direct effect makes communication more attractive, for both the DM andExpert 1. Second, the increase in ↵1 has an indirect effect on the DM through itseffect on Expert 1, and vice versa. Due to complementarity, an increase in one teammember’s effort raises the marginal productivity of the counterpart’s effort. Thedirect and indirect effects thus reinforce each other, so both r

⇤1 (↵1) and s

⇤1 (↵1) are

increasing in ↵1.When ↵1 > ↵

⇤, as ↵1 increases, the DM becomes more convinced that x1 = x, sothe marginal value of acquiring information decreases. Hence, r⇤1 (↵1) is decreasingin ↵1. From Lemma 1, we know that s

⇤1 (↵1) = 0 in this region.

At ↵

⇤, the DM’s default choice changes from not taking A1 to taking A1, so theexpert’s communication effort drops to zero. Due to complementarity, this lowersthe marginal benefit of the DM’s effort, so her attention drops discontinuously.

Claim A2.2 Fix the attractiveness of expert 2’s action, ↵2. The expected utilityof Sender 1 in equilibrium increases continuously with ↵1 for ↵1 2 (0,↵

⇤), increases

discontinuously at ↵

⇤, and increases continuously for ↵1 2 (↵

⇤, 1).

Proof of Claim A2.2 For any ↵1 2 (0, 1), an increase in ↵1 has a positive directeffect on the utility of Expert 1: for given effort levels on the part of Expert 1 andthe DM, an increase in ↵ raises the probability that trade will occur. In additionto this direct effect, an increase in ↵1 affects Expert 1 because the optimal effortschange. I show that this second effect reinforces the direct effect.

35

We start from ↵1 = ↵L < ↵

⇤, and the associated equilibrium (r

⇤1(↵L), s

⇤1(↵L), s

⇤2(↵L)) 2

(0, 1)

3 (for a given ↵2). I compare Expert 1’s expected utility in this equilibriumto that in an equilibrium where ↵1 = ↵H = ↵L + ", ↵H < ↵

⇤. The equilibriumassociated with ↵H , (r

⇤1(↵H), s

⇤1(↵H), s

⇤2(↵H)) 2 (0, 1)

3, satisfies r

⇤1(↵H) > r

⇤1(↵L)

and s

⇤1(↵H) > s

⇤1(↵L). When ↵1 ↵

⇤, for a given level of effort on the part ofExpert 1 , his expected utility is increasing in the attention that he gets from theDM. Thus, even if Expert 1’s effort were held fixed at s⇤1(↵L) when ↵1 = ↵H , Expert1 would be strictly better off getting attention r

⇤1(↵H) from the receiver than getting

attention r

⇤1(↵L) < r

⇤1(↵H). Clearly, then, Expert 1 is strictly better off in the equi-

librium associated with ↵H—where the DM devotes attention r

⇤1(↵H) to him and

he plays his best reply, s⇤1(↵H)—than in the equilibrium associated with ↵L. Hence,the expected utility of Expert 1 in equilibrium increases with ↵1 for ↵1 2 (0,↵

⇤).

Because all best reply functions and utility functions are continuous, the expectedutility increases continuously.

When ↵1 > ↵

⇤, his expected utility is decreasing in the attention that he getsfrom the DM. Sender 1’s effort is fixed at zero when ↵1 > ↵

⇤; and the DM’s attentionr

⇤1(↵1) is decreasing in ↵1. Thus, as ↵1 increases, Expert 1’s expected utility increases

because he gets less (undesirable) attention from the receiver. Because all bestreply functions and utility functions are continuous, the expected utility decreasescontinuously.

At ↵

⇤, Expert 1’s effort cost drops discontinuously (to zero); moreover, the at-tention he receives drops discontinuously as the DM’s decision rule changes from anopt-in to an opt-out rule. Both of these changes raise Expert 1’s expected utilitydiscontinuously.

Claim A2.3 When Expert 2 wants the DM’s attention (↵2 ↵

⇤), Expert 2’sexpected utility is a strictly decreasing function of the attention given to the otherexpert, r⇤1 (↵1).

Proof of Claim A2.3 This follows immediately from the facts that (i) US2 (↵1)

is increasing in r

⇤2 (↵1) for ↵2 ↵

⇤, and (ii) r

⇤2 (↵1) = 1 � r

⇤1 (↵1). Here, (i) follows

from Claims 1 and 2, and (ii) is the DM’s budget constraint.

Claim A2.4 When Expert 2 does not want the DM’s attention (↵2 > ↵

⇤), Expert2’s expected utility is a strictly increasing function of the attention given to the other

36

expert, r⇤1 (↵1).

Proof of Claim A2.4 This follows immediately from the facts that (i) US2 (↵1)

is decreasing in r

⇤2 (↵1) for ↵2 > ↵

⇤, and (ii) r

⇤2 (↵1) = 1� r

⇤1 (↵1). Here, (i) follows

from Claims 1 and 2, and (ii) is the DM’s budget constraint.

A.3 Proof of Corollary 1

This follows from Claims 1, 3, and 4 of the proof of Proposition 1.

A.4 Proof of Proposition 2

I first establish a preliminary result:

Lemma (Symmetric information outcome). Assume that the DM faces a cognitiveconstraint such that there exists a lower bound on the amount of (non-zero) attentionthat she can give to any one sender; ri 2 {0} [ [r, 1] for all i. Denote by t (r) thehighest number of senders that the DM can split her attention between if she splitsher attention equally among them, given the cognitive constraint r. Assume thatthere are N↵ high types (↵ = ↵) and N↵ low types (↵ = ↵), with t (r) < N↵ << N↵

and ↵ < ↵ < ↵

⇤. Under symmetric information, there is an (essentially unique)equilibrium in which the DM communicates with exactly t (r) high types.

I establish this Lemma in three steps.

Claim A4.1 Assume that there are N↵ � 2 identical experts with ↵ = ↵ < ↵

⇤.Then, there exists a unique equilibrium of this game, in which r

⇤i =

1N

↵

.

Proof of Claim A4.1 The problem of Si, i 2 {1, 2, ..., N↵} is characterized in theproof of Lemma 1, and Si’s unique best reply function is given by s

⇤i (bri) 2 (0, 1).

By symmetry, s⇤1 (·) = ... = s

⇤N

↵

(·) ⌘ s

⇤(·). The problem of the DM is given by

max

{r1,r2,...,rN�1}2[0,1](N↵

�1)

(x↵

p (s1, r1) + ...+ p

sN , 1�

i=N↵

�1X

i=1

ri

!!)

37

For given (conjectured) bs1, ..., bsN↵

, her first-order conditions yield

@p (bs1, r1)@r1

= ... =

@p (bsN↵

�1, rN↵

�1)

@rN↵

�1=

@p

✓bsN

↵

, 1�i=N

↵

�1Pi=1

ri

◆

@

✓1�

i=N↵

�1Pi=1

ri

◆ .

Substituting the experts’ best reply functions into this condition yields

@p (s

⇤(r1) , r1)

@r1= ... =

@p

✓s

⇤✓1�

i=N↵

�1Pi=1

ri

◆, 1�

i=N↵

�1Pi=1

ri

◆

@

✓1�

i=N↵

�1Pi=1

ri

◆ . (13)

Clearly, r⇤1 = r

⇤2 = r

⇤3 = ... = r

⇤N

↵

⌘ r

⇤ satisfies (13). Because the DM exhaustsher attention constraint in any equilibrium, there exists a unique symmetric equi-librium of this game, given by (r

⇤, s⇤) =

⇣1N

↵

, s⇤⇣

1N

↵

⌘⌘, where s⇤

⇣1N

↵

⌘is a vector

�s

⇤1, ..., s

⇤N

↵

�such that s

⇤i = s

⇤⇣

1N

↵

⌘for all i.

There exists no asymmetric interior equilibrium (where r

⇤i > 0 for all i and

r

⇤i 6= r

⇤j for some i, j such that i 6= j). To see this, define g(ri) ⌘ p2 (s

⇤(ri) , ri). By

the proof of Lemma 1, g (ri) is strictly increasing in ri. Hence, if r⇤i 6= r

⇤j for some

i, j such that i 6= j, (13) must be violated.There exists no equilibrium such that r

⇤i = 0 for some i. This follows directly

from (i) for all si 2 [0, 1] :

@p(si

,ri

)@r

i

> 0 for all ri 2 (0, 1), (ii) @p(si

,ri

)@r

i

! 0 as ri ! 1,and (iii) @p(s

i

,ri

)@r

i

! 1 as ri ! 0.Thus, the symmetric equilibrium is the unique equilibrium of this game.

Claim A4.2 Assume that the DM faces a cognitive constraint such that thereexists a lower bound on the amount of (non-zero) attention that she can give to anyone sender; ri 2 {0} [ [r, 1] for all i. Then, there exists a unique equilibrium of thisgame, in which r

⇤i =

1t(r) , where t (r) is the highest number of senders that the DM

can split her attention between, given the cognitive constraint r.

Proof of Claim A4.2 In the unconstrained optimum derived in the proof of Claim1, as N↵ increases, r⇤ =

1N

↵

⌘ r

⇤(N↵) decreases monotonically. Thus, there exists

some integer t (r) such that r

⇤(t (r)) > r > r

⇤(t (r) + 1). By the proof of Claim

1, the DM strictly prefers to communicate with t (r) senders over communicating

38

with strictly fewer senders. Because the DM exhausts her attention constraint inany optimum, there exists a unique symmetric equilibrium of this game, given by(r

⇤, s⇤) =

�1t , s

⇤ �1t

��, where s⇤

�1t

�is a vector (s

⇤1, ..., s

⇤t ) such that s

⇤i = s

⇤ �1t

�for

all i. This implies that the DM fares worse in an equilibrium where less than t hightype experts enter than in an equilibrium where t (or more) high types enter.

Claim A4.3 Assume that the DM faces a cognitive constraint r and that thereare N↵ high types (↵ = ↵) and N↵ low types (↵ = ↵), with t (r) < N↵ << N↵.Under symmetric information, the DM communicates with t (r) high types.

Proof of Claim A4.3 Because ↵ > ↵, and because the DM ’s expected utilityfrom communication with an expert is increasing in the expert’s type (↵), the DM ’sexpected utility from devoting attention r = 1/t (r) to a high type is higher than herexpected utility from devoting the same amount of attention to a low type. Becauset (r) < N↵, the DM only communicates with high types. By the proof of Claim 2,the DM communicates with exactly t (r) high types.

Having established the Lemma, the proof now proceeds in four steps.

Claim A4.4 When qS 2⇣q

S, qS

⌘, there exists a fully revealing equilibrium where

only high-quality experts approach the DM. She obtains the same expected decisionpayoff as under perfect information.

Proof of Claim A4.4 We derive conditions under which equilibria with cue com-munication exist. We postulate an equilibrium such that P high types send cues tothe DM , where t P N↵, zero low types send a cue to the DM , and the DM

devotes r

⇤t = 1/t to t experts chosen randomly among the P high types who send a

cue, where t ⌘ t (r), and zero attention to all other experts. In such an equilibrium,a low type refrains from sending a cue iff

qS > t

1

P + 1

d↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆� c(s

⇤↵

✓1

t

◆)

�. (14)

39

Exactly P high types send a cue iff

t

1P+1

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤< qS

< t

1P

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤

(15)The fact that s

⇤↵

�1t

�is a high type’s best reply implies that

d↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆� c

✓s

⇤↵

✓1

t

◆◆< d↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆� c

✓s

⇤↵

✓1

t

◆◆. (16)

By (16), (15) implies (14). Hence, if (15) is satisfied, the postulated equilibrium isincentive compatible for all experts. By (16), there exists a nonempty range of qSsuch that (15) is satisfied.

Because only high types send cues, an expert’s cue communication decision re-veals his type, so the DM need not assimilate the cues. By the proof of Claim 3,the DM ’s preferred attention allocation is to communicate with t high types. Sheis indifferent between the P high types who send cues to her. Thus, randomizingbetween all experts who send her a cue, and devoting zero attention to all expertswho do not, is incentive compatible for the DM . Hence, the postulated FRE existsif (15) holds.

The FRE is such that all N↵ high types send cues in equilibrium (but, if therewere N↵ + 1 high types, the last one would not enter) when qS satisfies

t

1N

↵

+1

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤< qS

< t

1N

↵

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤

(17)The left-hand side of (17) gives the expected utility from entry in the presence ofN↵ high types for a (hypothetical) (N↵ + 1)th high type. The expected utility fromentry in the presence of of N↵ high types is strictly smaller for a low type. Thus,denoting this expected utility by q

S, we have that

q

S< t

1

N↵ + 1

d↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆� c

✓s

⇤↵

✓1

t

◆◆�.

There exists a the FRE is such that all N↵ high types send cues in equilibrium (but

40

no low types) when qS satisfies

q

S< qS < t

1

N↵

d↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆� c

✓s

⇤↵

✓1

t

◆◆�. (18)

The FRE is such that exactly t high types send cues in equilibrium (but no lowtypes) when qS satisfies

t

1t+1

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤< qS

<

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤

Thus, FRE in which t or more high types (but no low types) send cues to the DM

exist iff qS > q

Sand qS <

⇥d↵p

�s

⇤↵

�1t

�,

1t

�� c

�s

⇤↵

�1t

��⇤⌘ qS . The DM ’s expected

utility, in any of these equilibria, is given by U

⇤R = x↵tp

�s

⇤↵

�1t

�,

1t

�, which is the

same decision payoff as she obtains in the perfect information case.

Claim A4.5 When qS falls below q

S, low quality experts also approach the re-

ceiver. She must either accept a lower expected decision payoff, or intensify hersearch for high-quality senders. In either case, her expected utility is strictly lowerthan when qS 2

⇣q

S, qS

⌘.

Proof of Claim A4.5 We first show that when qS falls below q

S, there exist only

equilibria that make the DM strictly worse off than when qS 2⇣q

S, qS

⌘.

When qS falls below q

S, (18) is violated. Thus, in any equilibrium with cue

communication, at least one low type sends a cue (and all high types). In such anequilibrium, the DM either (i) opens zero cues, but randomly chooses to communi-cate with t experts, or (ii) opens at least one cue. We show that both of these maybe consistent with equilibrium play. We show this in the context of an equilibriumin which exactly one low type sends a cue.

If the DM plays strategy (i), the equilibrium must satisfy

tN

↵

+2

⇥s↵p

�x

⇤↵

�1t

�,

1t

�� c

�x

⇤↵

�1t

��⇤< qS

<

tN

↵

+1

⇥s↵p

�x

⇤↵

�1t

�,

1t

�� c

�x

⇤↵

�1t

��⇤

where we use the fact that the DM will devote the same amount of attention toevery expert with whom she communicates (as the experts’ types are their privateinformation). We denote by U

ranR (N↵, 1, t) the DM ’s ex ante utility in this random-

41

ization equilibrium where N↵ high types and one low type send cues to the DM whochooses t experts among them randomly. We have that

U

ranR (N↵, 1, t) = xµ (N↵, 1, t)

↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆+ (t� 1)↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆�

+ x↵ (1� µ (N↵, 1, t)) tp

✓s

⇤↵

✓1

t

◆,

1

t

◆

where µ (N↵, 1, t) is the probability that the (only) low type is among the t ex-perts that the DM randomly picks from the N↵ + 1 available experts. Becauseµ (N↵, 1, t) > 0, U ran

R (N↵, 1, t) < U

⇤R = x↵tp

�s

⇤↵

�1t

�,

1t

�.

If the DM instead plays strategy (ii), and if she commits to opening exactly onecue, her expected utility satisfies

U

cueR (N↵, 1, t) � �qR +

1

N↵ + 1

x↵tp

✓x

⇤↵

✓1

t

◆,

1

t

◆

+

✓N↵

N↵ + 1

◆✓x↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆+ U

ranR (N↵ � 1, 1, t� 1)

◆

With probability 1N

↵

+1 , the DM opens the cue sent by the (only) low type, inwhich case she communicates with the t high types and gets her preferred attentionallocation. With probability N

↵

N↵

+1 , the cue was sent by a high type, so the DM

communicates with this expert and randomly picks (t � 1) others. In this case,her ex ante expected decision utility is greater than or equal to x↵p

�s

⇤↵

�1t

�,

1t

�+

U

ranR (N↵ � 1, 1, t� 1) (where the inequality is strict if the DM chooses to treat

the identified high type preferentially, at the expense of dropping one expert ofunknown type). Because the DM does not obtain her preferred attention allocationwith probability one, U cue

R (N↵, 1) < U

⇤R.

We now show that both (i) and (ii) may, depending on the parameter values, bepreferred by the DM :

In expectation, choosing t experts at random among N↵+1 is strictly worse thanobserving one high type and choosing the other (t� 1) at random, i.e.,

1

N↵ + 1

x↵tp

✓x

⇤↵

✓1

t

◆,

1

t

◆+

N↵

N↵ + 1

✓x↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆+ U

ranR (N↵ � 1, 1, t� 1)

◆

� U

ranR (N↵, 1, t) > 0 (19)

42

Equation (19) implies that there exists a non-empty range of qR such thatU

cueR (N↵, 1, t) > U

ranR (N↵, 1, t), given by

1

N↵ + 1

x↵tp

✓x

⇤↵

✓1

t

◆,

1

t

◆+

N↵

N↵ + 1

✓x↵p

✓s

⇤↵

✓1

t

◆,

1

t

◆+ U

ranR (N↵ � 1, 1, t� 1)

◆

� U

ranR (N↵, 1, t) � qR. (20)

We now note that when the DM prefers a strategy in which she commits to openingexactly one cue to a strategy in which she randomizes, the DM also prefers a strat-egy in which she opens at least one cue to randomization. Thus, whenever (20) issatisfied, the DM opens at least one cue. Because the left-hand side of (20) is finite,the reverse is true for large enough qR, i.e., U cue

R (N↵, 1, t) < U

ranR (N↵, 1, t).

Claim A4.6 When qS ! 0, the number of low quality experts who approachthe DM becomes so large that she ceases to screen experts for quality. The DM’sexpected decision payoff is strictly smaller than that obtained in any equilibriumwhere cue communication takes place.

Proof of Claim A4.6 Suppose that N↵ is infinite. As qS ! 0, the number of lowtypes that wish to send a cue to the DM , n, approaches infinity, which implies thatN

↵

N↵

+n ! 1 and N↵

N↵

+n ! 0. Thus, U cueR (N↵, n, t) ! x↵tp

�x

⇤↵

�1t

�,

1t

�� qR. Because

U

ranR (N↵, n, t) ! x↵tp

�x

⇤↵

�1t

�,

1t

�, the DM strictly prefers not to open any cue in

the limit. In an equilibrium in which she randomizes, she is worse off, the largerthe share of low type experts. Thus, she is clearly worse off than in any equilibriumwhere she reads cues.

Claim A4.7 The decrease in the DM’s expected utility is monotonic for qS < q

S.

Proof of Claim A4.7 When the DM randomly chooses experts, this follows di-rectly from the fact that N

↵

N↵

+n and N↵

N↵

+n change monotonically with the number ofentering low types, n. So long as the DM opens cues, the value of opening one cueis decreasing in N

↵

N↵

+n , which is monotonically increasing in n.

43

A.5 Proof of Proposition 3

Claim A5.1 If the DM assimilates one cue, she continues to assimilate cues untilshe identifies the relevant topic.

Proof of Claim A5.1 Suppose that the DM has launched k topics. Consider thefirst cue that the DM assimilates. She incurs the cost qR. With probability 1/k,she finds the relevant topic, and devotes all of her attention to this topic. Withprobability (k� 1)/k she does not find the relevant topic. In this situation, the DMalways assimilates a second cue: the cost of assimilation is still qR; however, theprobability that she identifies the relevant topic is 1/(k � 1) > 1/k. Hence, if theDM assimilated the first cue, she assimilates a second cue in the even that the firsttopic is irrelevant. Repeating this argument yields that she, if she assimilates onecue, continues to assimilate cues until she finds the relevant topic.

Claim A5.2 There exists a number of cues (topics) k⇤ such that, if the DM obtainsmore than k

⇤ topics, then she assimilates no cue. Instead, she randomly chooses t

topics that she divides her attention between (equally) in the deliberation stage.

Proof of Claim A5.2 Consider the DM’s expected utility if she assimilates cues.If the first cue that she assimilates is the relevant one, which happens with probability1/k, then her expected payoff is (⇡ � qR), where ⇡ = ↵x+(1�↵)x�p(0, 1)(1�↵)x.

That is, her expected payoff is the expected payoff from the action, adjusted for thefact that she may find out, through her information acquisition on the relevant topic,that the product quality is low (and opt out). When she devotes all of her attentionto this topic, and the expert devotes zero effort, the probability that she obtainssuch information is given by p(0, 1) in the event that the product quality indeed islow, which happens with probability (1� ↵). If the first cue that she assimilates isnot the relevant one, which happens with probability (k� 1)/k, then she assimilatesa second cue.

If the second cue that she assimilates is the relevant one, which happens withprobability 1/(k � 1), then her expected payoff is (⇡ � 2qR). If the second cue isnot the relevant one, then she continues. Repeating this argument yields that her

44

expected payoff from assimilating cues (until she finds the relevant one) is given by

1

k

(⇡ � qR) +(k � 1)

k

1

(k � 1)

(⇡ � 2qR) +(k � 1)

k

(k � 2)

(k � 1)

1

(k � 2)

(⇡ � 3qR)+

...+

1

k

(⇡ � kqR) =1

k

i=kX

i=1

(⇡ � iqR) = ⇡ � qR

k

(1 + 2 + ...+ k) =

⇡ � qR

k

k (1 + k)

2

= ⇡ � qR(1 + k)

2

.

Because qR(1+k)

2 increases in k without bound, there exists a k

⇤ such that

⇡ � qR(1 + k

⇤)

2

> 0 > ⇡ � qR(1 + (k

⇤+ 1))

2

.

If the DM does not assimilate any cue, but instead randomly chooses t out ofthe k cues available to her, her expected utility is given by ⇡

0= ↵x + (1 � ↵)x �

tkp(0,

1t )(1�↵)x, since she chooses t/k out of the topics available, and hence picks the

relevant topic with probability t/k. Among the t topics that she randomly chooses,she devotes 1/t of her attention to each of them. Because ↵x+(1�↵)x > 0, we havethat ⇡

0> 0. Clearly, the DM strictly prefers to randomize over assimilating cues if

the expert makes more than k

⇤ topics available. The DM prefers to randomize whenher expected payoff from randomization exceeds her expected payoff from openingcues, i.e., when

↵x+ (1� ↵)x� t

k

p(0,

1

t

)(1� ↵)x > ↵x+ (1� ↵)x� p(0, 1)(1� ↵)x� qR(1 + k)

2

.

We know that this holds when k > k

⇤. We denote the smallest number of topicssuch that the DM prefers to randomize by k

⇤⇤. Clearly, k⇤⇤ k

⇤.

Claim A5.3 The expert either launches only one topic, or at least k

⇤⇤ topics. IfqS is small, he launches at least k

⇤⇤ topics.

Proof of Claim A5.3 If the expert launches only one topic (the relevant one),then the DM devotes all of her attention to this topic. Thus, she opts out withprobability p(0, 1)(1� ↵).

If he launches more than one, but fewer than k

⇤⇤ topics, the DM assimilatescues until she finds the relevant topic. Then, she devotes all of her attention to this

45

topic. Hence, she opts out with the same probability; however, the expert incurreda higher cost of making the (additional) topics available. Thus, the expert strictlyprefers launching one topic to launching strictly more than one, but fewer than k

⇤⇤,topics.

If he launches at least k

⇤⇤ topics, the DM randomly chooses t out of the k

⇤⇤

topics, and devotes attention 1/t to each of the selected topics. In this case, she optsout with probability t

kp(0,1t )(1�↵) < p(0, 1)(1�↵). Clearly, if the cost of launching

a topic, qS , is small enough, the expert strictly prefers to launch at least k

⇤⇤ topics.

Claim A5.4 When qS = 0, the mandate to disclose the relevant topic has no effecton the DM’s expected utility; she does not process the relevant information at all.

Proof of Claim A5.4 When qS ! 1, the number k of topics launched goes to1, and the probability that the DM opts out goes to limk!1

⇥tkp(0,

1t )(1� ↵)

⇤= 0.

Hence, the mandate to disclose the relevant topic has no effect on the DM’s expectedutility; the expected utility is simply given by ↵x+ (1� ↵)x, which is her expectedutility in the absence of any mandate.

46